# Inertia mismatch (ServoSoft)

#### What is inertia mismatch

It is the ratio of system load inertia to motor rotor inertia. It is denoted by λ and it is the main indicator of the application’s controllability.
For dynamic systems, the larger λ is, the harder the system is to control.

Also good to remember that controllability and observability are dual motions which comes in handy when identifying unknown systems.

#### Why is it important

Inertia is the resistance of any physical object to any change in its speed. Objects with greater mass have more inertia. It takes more force to change its motion.

Quick speed changes and positioning become very difficult if load inertia is high compared to the motor’s rotor inertia - inertia ratio λ is high.

High load-to-motor inertia ratios decrease the operating bandwidth of the machine, make the motor work harder than it should, and often lead to increased settling times, resulting in decreased overall performance.

#### Reducing IM

One way to reduce the inertia ratio is to use gearboxes since gear ratio has an inverse square effect on the inertia of the load. This enables higher-speed operations and/or smaller and cheaper motors. Another approach is to switch to a bigger motor with larger inertia, or you can use a direct drive motor without bearings or increase the rigidity of the mechanical connection of the system.

##### Mechanical system

Mechanical changes to the system can have a significant effect on performance. Consider an axis driving a wheel with a 20:1 inertia ratio. Now increase the diameter of the wheel. Even if the mass remains the same, the inertia will rise. The customer makes a request and you make a few small changes and now your inertia mismatch jumps to 50:1. All of a sudden you get sluggish control.

##### Typical values of inertia mismatch

Keep in mind that there is no general formula for calculating ideal inertia ratio – it has to be calculated for every individual system, but here are some typical values depending on your application.

#### Sources:

B&R:

ServoSoft

ServoSoft - basics (webinar)

ServoSoft - online course

SRL465-Drivetrain sizing

(requires logging in to the B&R website)

Web:

Drivetrain sizing and inertia mismatch

Machine Design Mistakes

Understanding the Mysteries of Inertia Mismatch

(both are really good reading material)

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Awesome, thank you for sharing @fikejja1!

Brilliant article. Do you know where is the limit of inertia mismatch in case of big winding machines like industrial printers? In the case of moving tons of paper or other material there is always discussion about inertia mismatch, because huge motors generates huge costs. When it comes to my experience - for velocity / torque controlled axes 30:1 is great, 50:1 is still ok, but 100:1 was terrible and axis tuning was not possible.

Hello Michal,

great question, but very difficult to answer I wouldn’t say that there is a “limit” for winding machines per se. Of course having the ratio at 30:1 is great, but you could have a ratio of 1000:1 - or more - for some less dynamic systems and still be able to tune to your specs - of course the requirements for overshoot, accel/decel limits will be lower.

However that is not always acceptable or desired…all depending on the specific machine

(keep in mind that every part of the system plays a role, e.g. for short and stiff shafts you can raise the IM and get away with it).

Excuse me, if this answer is too vague. I will write some general article about tuning and control theory in the future as a quasi guide for future reference.

Best
Jan

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